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61

Sep 22, 2013
09/13

by
Lars Allermann; Johannes Rau

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We introduce an improved version of rational equivalence in tropical intersection theory which can be seen as a replacement of chapter 8 of our previous article arXiv:0709.3705v2. Using this new definition, rational equivalence is compatible with push-forwards of cycles. Moreover, we prove that every tropical cycle in R^r is equivalent to a uniquely determined affine cycle, called its degree.

Source: http://arxiv.org/abs/0811.2860v2

3
3.0

Jun 30, 2018
06/18

by
Hannah Markwig; Johannes Rau

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In this paper, we define tropical analogues of real Hurwitz numbers, i.e. numbers of covers of surfaces with compatible involutions satisfying prescribed ramification properties. We prove a correspondence theorem stating the equality of the tropical numbers with their real counterparts. We apply this theorem to the case of double Hurwitz numbers (which generalizes our result from arXiv:1409.8095).

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1412.4235

3
3.0

Jun 30, 2018
06/18

by
Lars Allermann; Simon Hampe; Johannes Rau

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This article discusses the concept of rational equivalence in tropical geometry (and replaces the older and imperfect version arXiv:0811.2860). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the "bounded" Chow groups of $\mathbb{R}^n$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest: We show that every tropical cycle in...

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1408.1537

3
3.0

Jun 29, 2018
06/18

by
Magdalena Lippa; Stefan Gillessen; Nicolas Blind; Yipting Kok; Senol Yazici; Johannes Weber; Oliver Pfuhl; Marcus Haug; Stefan Kellner; Ekkehard Wieprecht; Frank Eisenhauer; Reinhard Genzel; Oliver Hans; Frank Haussmann; David Huber; Tobias Kratschmann; Thomas Ott; Markus Plattner; Christian Rau; Eckhard Sturm; Idel Waisberg; Erich Wiezorrek; Guy Perrin; Karine Perraut; Wolfgang Brandner; Christian Straubmeier; Antonio Amorim

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The VLTI instrument GRAVITY combines the beams from four telescopes and provides phase-referenced imaging as well as precision-astrometry of order 10 microarcseconds by observing two celestial objects in dual-field mode. Their angular separation can be determined from their differential OPD (dOPD) when the internal dOPDs in the interferometer are known. Here, we present the general overview of the novel metrology system which performs these measurements. The metrology consists of a three-beam...

Topics: Astrophysics, Instrumentation and Methods for Astrophysics

Source: http://arxiv.org/abs/1608.04888

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30

Sep 21, 2013
09/13

by
Lars Allermann; Johannes Rau

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We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in R^n and then for "abstract" cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in R^n.

Source: http://arxiv.org/abs/0709.3705v3

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58

Sep 23, 2013
09/13

by
Georges Francois; Johannes Rau

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We define an intersection product of tropical cycles on matroid varieties (via cutting out the diagonal) and show that it is well-behaved. In particular, this enables us to intersect cycles on moduli spaces of tropical rational marked curves $M_{0,n}$ and $M_{0,n}(\R^r, d)$. This intersection product can be extended to smooth varieties (whose local models are matroid varieties). We also study pull-backs of cycles and rational equivalence.

Source: http://arxiv.org/abs/1012.3260v1

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331

Apr 30, 2012
04/12

by
Lowth, Robert, 1710-1787; Sicard, M; Rau, Sebald Tulco Johannes, 1765-1807

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Topics: Bible, Poésie hébraïque

8
8.0

Jun 28, 2018
06/18

by
Ilia Itenberg; Grigory Mikhalkin; Johannes Rau

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From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in $\mathbb{RP}^2$ in the spirit of Hilbert's 16th problem.

Topics: Mathematics, Algebraic Geometry

Source: http://arxiv.org/abs/1509.05228

32
32

Mar 26, 2019
03/19

by
Rau, Johannes Jacobus, 1668-1719

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4 unnumbered pages, 39 pages, 1 unnumbered page ; 4to (23 cm)

Topics: Human anatomy, Human anatomy

7
7.0

Jun 30, 2018
06/18

by
Mathieu Guay-Paquet; Hannah Markwig; Johannes Rau

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We investigate the combinatorics of real double Hurwitz numbers with real positive branch points using the symmetric group. Our main focus is twofold. First, we prove correspondence theorems relating these numbers to counts of tropical real covers and study the structure of real double Hurwitz numbers with the help of the tropical count. Second, we express the numbers as counts of paths in a subgraph of the Cayley graph of the symmetric group. By restricting to real double Hurwitz numbers with...

Topics: Mathematics, Combinatorics, Algebraic Geometry

Source: http://arxiv.org/abs/1409.8095

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56

Sep 18, 2013
09/13

by
Hannah Markwig; Johannes Rau

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We define tropical Psi-classes on the moduli space of rational tropical curves in R^2 and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi- and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin's lattice path algorithm and counts rational plane tropical curves...

Source: http://arxiv.org/abs/0809.1102v2

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38

Sep 22, 2013
09/13

by
Johannes Rau

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This article tries to answer the question: How far can the algebro-geometric theory of rational descendant Gromov-Witten invariants be carried over to the tropical world? Given the fact that our moduli spaces are non-compact, the answer is surprisingly positive: We discuss universal families and the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we give two criteria that suffice to prove the tropical version...

Source: http://arxiv.org/abs/0812.3678v1

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40

Feb 21, 2019
02/19

by
Rau, Johannes Jacobus, 1668-1719

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16 pages, 1 unnumbered leaf, 6 pages : (4to)

Topic: Ruysch, Frederik, 1638-1731

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47

Feb 21, 2019
02/19

by
Rau, Johannes Jacobus, 1668-1719

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16 pages, 1 unnumbered leaf, 6 pages : (4to)

Topic: Ruysch, Frederik, 1638-1731

380
380

May 1, 2012
05/12

by
Lowth, Robert, 1710-1787; Sicard, M; Rau, Sebald Tulco Johannes, 1765-1807

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Topics: Bible, Poésie hébraïque